Born (Gap-14/15) — Born rule: the gate anchor ledger

The honest one-line: the Born rule has a real, checkable negative win — an explicit countermodel proving the frozen framework's own gauge (BRST) axioms do not force non-contextuality, so the program imports the load-bearing hypothesis on the record rather than smuggling it — and on the ratified board (2026-07-08) the gate stands CERTIFIED-IRREDUCIBLE (REDUCED-TO-AXIOM on the named posit BORN-A1) · RESOLVED +0, with the residual family shown openly below: Leg A1 (the functional form) rides an import-forbidden algebra type as a named, value-free posit above the measured quantum-kinematics floor, and Leg A2 (the selector measure) carries its uniqueness-fails residual in the open.

This page is the gate-specific instantiation of the anchoring method: it takes the master anchor and the four bridges and applies them, object by object, to one gate. Every exact thing the Born gate touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. It follows the same eleven-part shape and the same universal table as the canonical SG-4 ledger.

The Born gate is not shaped like an anomaly-ledger gate. There is no exact rational obstruction that vanishes on the frozen spectrum; instead the genuine win is a proven separation (a no-go countermodel) plus a measure-existence localization. The page adapts honestly to that nature.

1. Gate status header

• Gate-level Born (Gap-14/15) roll-up (ratified board 2026-07-08): CERTIFIED-IRREDUCIBLE · RESOLVED +0 on the named posit BORN-A1. The earlier OPEN roll-up label from the superseded grading rule is preserved below as history.
• Taxonomy reconciliation (2026-07-05; ratified 2026-07-08): under the ratified closure taxonomy the gate-level grading is CERTIFIED-IRREDUCIBLE (REDUCED-TO-AXIOM on the named posit BORN-A1 non-contextuality) · RESOLVED +0, read as TERMINAL + RESIDUALS-SHOWN — a reached certified-irreducible terminal (the named, value-free non-contextuality posit BORN-A1, certified irreducible on the measured quantum-kinematics floor) with its residual family shown, not a physics-closed claim. The residual family listed in this ledger (Leg A2 uniqueness and Holes A–I) remains listed and carried unchanged, as residuals shown above the axioms. The facts are unchanged: the historical OPEN roll-up and the per-leg OPEN / AXIOM-OPEN labels on the rows below reflect the superseded least-closed-residual grading rule, kept as the residual record — not different facts; the closure-of-record dossier carries the grade of record.
• The two legs, graded apart (residual record; per-leg labels from the superseded pass, facts unchanged):
  • Leg A1 — the functional form $\rho\mapsto\mathrm{Tr}(\rho E)$ (the quadratic $|\psi|^2$ weight on rays of a fixed Hilbert space, Gap-14, intra-sector). AXIOM-OPEN (imported, import-forbidden). The Born functional form itself stays OPEN.
  • Leg A2 — the selector measure $\mu$ on the state space (the inter-sector / superselection weights, Gap-15). OPEN — uniqueness fails, the point-mass input is disclosed-not-derived, no clean inter-sector invariant.
• The closed local leg (a proven no-go, not a derivation of the rule): given the BRST physical algebra, the bare gauge axioms force gauge-orbit equivariance but not non-contextuality: $$ (G1)+(G2)\;\not\Rightarrow\;\text{non-contextuality}. $$ This is DERIVED as a separation theorem — checkable, terminal as a no-go — and it is the gate's real win. It does not close the gate; it measures the size of the import the gate must pay.

Status, split so it cannot be misread: - The A1 countermodel (separation): DERIVED — a pencil-checkable witness; gauge axioms admit a gauge-invariant, orbit-equivariant, contextual valuation. - Leg A1 functional form (BORN-A1): CERTIFIED-IRREDUCIBLE (named value-free posit, ratified 2026-07-08) — imported on the record; superseded per-leg label: AXIOM-OPEN / import-forbidden. - Leg A2 measure (BORN-A2): OPEN — existence recovered on a compact chamber, uniqueness provably fails. - The measured floor (ANCHOR-BORN-QUANTUM-KINEMATICS): MEASURED-ANCHOR — the legitimate $\geq 1$ floor, not a closure.

A grade-discipline note: an earlier reading graded A1 "AXIOM-CLOSED." The post-atomicity reading re-grades A1 to AXIOM-OPEN / import-forbidden on the principle ANCHORED ≠ DERIVED; AXIOM-CLOSED ≠ atomic — a leg is terminal only if its posit is atomic (proven irreducible or directly measured), and BORN-A1 was, at that pass, neither. Superseded on this point by the 2026-07-08 ratification: the closure-of-record dossier certifies BORN-A1 irreducible — all four discharge routes exhausted — and fixes the grade CERTIFIED-IRREDUCIBLE (REDUCED-TO-AXIOM on BORN-A1) · RESOLVED +0; the note above is kept as the pre-certification record.

selection ≠ derivation · given-E ≠ derivation-of-E · frozen/reproducible ≠ proven-unique.

2. Frozen inputs (what the Born gate stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 / manifest meta a5b1e6f9d951 (K₆ = SU(3)/T² coset + its admissibility-chamber structure). The branch is READ-ONLY. These hashes are audit anchors — they certify which object was tested and that it cannot be quietly retuned. They do not validate the physics.
• The measured floor ANCHOR-BORN-QUANTUM-KINEMATICS: a Hilbert space plus the observed probability law $p(E)=\mathrm{Tr}(\rho E)$, taken as the measured floor the reduction sits within. This is MEASURED-ANCHOR / charged — the gate does not derive quantum kinematics; every "sharpens" below is a statement given this floor.
• The cascade object $a_6$ (Gap-01's signed-total heat-kernel coefficient): GIVEN / upstream-charged, and for the Born decoherence channel not yet deliverable. The gate does not own it.

3. The object anchors (given / upstream)

The three distinct measure-theoretic objects the gate tracks apart, never conflated:

The BRST physical space $\mathcal H_{\rm phys}=\ker Q_{\rm BRST}/\operatorname{im}Q_{\rm BRST}$ and its gauge-invariant observable algebra $\mathcal A$ are the structures Leg A1 acts on. The frozen branch supplies $\dim\mathcal H_{\rm phys}\geq 3$ in surplus. Status: GIVEN / upstream-inherited — not Born-derived.

4. Root and master-anchor traceability

Deep roots that are load-bearing for the Born gate:

Scale and causal order are not primary load-bearing anchors for the Born legs; the granularity/cost-floor lever is explicitly wrong-shape here and is recorded as an honest negative, not a hidden key.

Master anchors in play: the measured floor (≥1 anchor) · no unpaid measure · the frozen branch · given-quantum-kinematics · the declared named posits BORN-A1 / BORN-A2 · open-residual discipline · the κ³/π falsification test (no reverse-engineered from the measured value measure).

5. The Born anchor ledger (the universal table)

6. The construction — the two legs, in full

6.1 Leg A1 — what BRST genuinely forces, and the separation

Fix the gauge-invariant observable von Neumann algebra $\mathcal A$ on $\mathcal H_{\rm phys}$. A frame function is a map $p:\mathcal P(\mathcal A)\to[0,1]$ with $p(I)=1$. BRST cohomology + gauge invariance impose exactly two things, both genuinely supplied by the frozen structure:

• $(G1)$ Admissibility. Only $Q_{\rm BRST}$-closed, gauge-invariant effects are observables — this restricts the domain of $p$ to $\mathcal P(\mathcal A)$; it says nothing about behavior across contexts.
• $(G2)$ Gauge-orbit equivariance. $p(E)=p(g\cdot E)$ — $p$ is constant along gauge orbits, forced because gauge-equivalent configurations are physically identical.

Given non-contextuality, the form $\mathrm{Tr}(\rho E)$ follows by a genuine theorem, with the theorem fixed by the algebra:

Three precision caveats, stated not glossed: (a) Mackey–Gleason yields a bounded linear functional; positivity and normalization are extra, and a normal state needs countable additivity (Yeadon). (b) "Born by fiat" is wrong in both directions — the theorem genuinely derives the quadratic form from additivity (more than fiat) but does not derive non-contextuality (less than a closure): hence one named axiom, not zero. (c) Theorem choice is premise-dependent — a type-I₂ summand blocks Mackey–Gleason.

The separation (the real, checkable win) ∎. There exists a valuation $p$ on $\mathcal P(\mathcal A)$ satisfying $(G1)$ and $(G2)$ exactly — fully gauge-invariant, fully orbit-equivariant — yet contextual, needing only $\dim\mathcal H_{\rm phys}\geq 3$, no type-I₂ subtlety, no Kochen–Specker coloring. Work in a 3-dimensional gauge-invariant block $B(\mathbb C^3)\subset\mathcal A$ with gauge acting trivially (the generic post-BRST situation), so $(G2)$ holds vacuously. Take two maximal resolutions of $I$ sharing the rank-1 projection $P_1=|1\rangle\langle 1|$: $$ \mathcal C_A:\; I = P_1 + |2\rangle\langle 2| + |3\rangle\langle 3|,\qquad \mathcal C_B:\; I = P_1 + |u\rangle\langle u| + |v\rangle\langle v|, $$ with $\{|u\rangle,|v\rangle\}$ any other ON basis of $\operatorname{span}\{|2\rangle,|3\rangle\}$. Assign context-label-first weights $$ p(P_1\mid\mathcal C_A)=a,\quad p(P_1\mid\mathcal C_B)=a',\qquad a\neq a', $$ each context internally normalized and non-negative. Then $(G1)$ holds (every $P_i$ is a genuine projection in $B(\mathbb C^3)\subset\mathcal A$), $(G2)$ holds exactly (gauge trivial), yet $p$ violates non-contextuality by construction (the shared $P_1$ carries different weight in the two contexts). $\blacksquare$

Diagnostic — the separation is specific, not vacuous. The two contexts $\mathcal C_A,\mathcal C_B$ are related by a unitary rotation in the $(2,3)$-plane that is not a gauge transformation (it mixes physical observables), so $(G2)$ cannot identify them and $a\neq a'$ survives. Any derivation reaching $\mathrm{Tr}(\rho E)$ from $(G1)+(G2)$ must silently impose $a=a'$ across contexts — and that imposition is non-contextuality, is Gleason's hypothesis, is the Born rule minus the exponent. BRST supplies the constraints; it does not supply the gluing. The gluing is BORN-A1.

6.2 Leg A2 — measure existence yes, uniqueness no

Leg A2 asks for a forced measure on the state space. The three standard obstructions all verify correct: 1. No $U(\mathcal H)$-invariant probability measure on infinite-dim $P(\mathcal H)$ (the unit sphere of an infinite-dim $\mathcal H$ is non-compact; F. Riesz's lemma). 2. Within-sector invariance does not fix inter-sector weights — for $S=\bigcup_a S_a$, every convex combination $\sum_a c_a\mu_a$ is $G$-invariant; invariance pins the measure only under ergodicity, which a nontrivial invariant decomposition is precisely the failure of. 3. Category mismatch: a Haar measure on a group/manifold is not the normalized effect-additive functional $\mu:E(H)\to[0,1]$; without explicit transport the chamber route can at best complement, never replace, the within-Hilbert functional.

The localization (the gain). On infinite-dim $P(\mathcal H)$ there is no invariant probability measure; on the compact chamber it exists (normalized Lebesgue on a box). The frozen branch's concrete candidate is the K₆ = SU(3)/T² Weyl chamber $$ C=[\tfrac12,\tfrac32]^3,\qquad W=S_3\;(|W|=6),\qquad \text{witness}\;(1,1,1). $$

Where it blocks. T7's uniqueness engine is the homogeneous-space theorem (a compact group acting transitively on $X$ has a unique invariant probability measure). The chamber's residual symmetry is the finite group $W=S_3$: 1. No transitivity — $|S_3|=6$; the orbit of any $u\in C$ has $\leq 6$ points, $C$ is an uncountable 3-dim continuum; a finite group cannot act transitively on a positive-dimensional space. 2. No ergodicity $\Rightarrow$ no uniqueness — the invariant measures are exactly $$ \mu=\tfrac16\sum_{\sigma\in S_3}\sigma_*(f\cdot\mathrm{Leb}),\qquad f\geq 0,\ \textstyle\int f=1, $$ an infinite-dimensional convex set. 3. The fixed locus is a line, not a point — $\mathrm{Fix}(S_3)=\{u_1=u_2=u_3\}$ is 1-dimensional; any probability measure on the diagonal is $S_3$-fixed, so even "$\delta$ at $(1,1,1)$" is a choice. By Weyl-rigidity $(1,1,1)$ is forced to be a critical point of any $S_3$-invariant density — distinguished, but not unique.

Net. A2 collapses to a single, finite-dimensional choice problem — which $S_3$-invariant density $f$ on $C$ is the physical one? — sharper than "posit a measure on an infinite-dim space," but still a selection, not a derivation. Naming the posit does not reduce the axiom count.

6.3 The which-outcomes leg — the honest halt

Decoherence (S2) and the system–bath split (S3) can at best select a pointer basis (the which-outcomes question); they do not deliver the weights (the with-what-probability question, S4). The one in-geometry channel, CH-2 (the heavy KK tower as a monitoring environment), raises NotImplementedError — a deliberate refusal to fabricate — because it cascades on Gap-01's $a_6$ object. No value, sign, or rate of CH-2 is asserted anywhere. The pointer-basis structural-consistency check S3.b carries the gate's self-demoting falsifier: a structurally inconsistent pointer basis turns Gap-14 into a decision-grade open blocker. The basis is not yet exhibited; the falsifier is neither fired nor cleared.

7. Declared-structure splits — why two axioms, not one

The single phrase "the Born rule" hides two distinct objects with two statuses, and the proposal to collapse them into one axiom was corrected to DUAL, not identical:

1. Leg A1 (intra-sector) — once non-contextuality is granted, the within-sector density operator is forced and unique (modulo the I₂ guard). The obstruction is the cross-context gluing $a=a'$. Status: AXIOM-OPEN / import-forbidden.
2. Leg A2 (inter-sector) — Gleason / Bunce–Wright is structurally silent across sectors: additivity gives a convex combination but pins no weights. Status: OPEN.

They belong to the same family ("a symmetry/consistency constraint fixes structure WITHIN an equivalence class but leaves a free parameter ACROSS classes") but live on different objects and cannot be collapsed into one axiom. Two axioms are required precisely because Gleason / Bunce–Wright closes the intra-block problem and is structurally silent on the inter-block problem.

A second split, inside A1: the vN-algebra type (no-I₂) is its own object, IMPORT-DEPENDENT — $Z(\mathcal A)$ is uncomputed and $\xi_{R4}$ is UNKNOWN, the two exact places a degree-2 central fiber could hide (I2-DANGER-OPEN). This is independent of non-contextuality: the §6.1 countermodel lives in an honest type-I₃ block where the type is settled and non-contextuality still fails.

8. The κ³/π falsification test — guard against true-by-construction

A proposed axiom or measure counts only if it would be written without knowing the Born rule is the target. Both named posits pass as statements: BORN-A1 is the standard non-contextuality hypothesis — value-free, identical to the independently-stated Mackey–Gleason hypothesis, no exponent 2, no overlap; BORN-A2 names the maximal-symmetry member of the invariant family — a symmetry-selection rule carrying no Born value. The disposition would be identical if the empirical Born exponent were $1.7$ or $3$: the countermodel and the chamber floor do not depend on the exponent.

The trap, applied to every hole in §10: a measure reverse-engineered so the selector reproduces the Born weights is true by construction and relocates the mystery rather than closing it. The cautionary precedent is the κ³/π / $a:=4\kappa/\sqrt 3$ case elsewhere in the corpus, where a reverse-engineered from the measured value number landed in a pre-registered window purely because it was reverse-engineered to.

9. The witness ledger — what reproduces, plainly

What does NOT reproduce: the Born functional form does not fall out (the chain geometry → non-contextuality → $\mathrm{Tr}(\rho E)$ is broken at the first link); the canonical measure is not paid (uniqueness fails, point-mass disclosed, inter-sector weights have no clean invariant); the §19B exhibit is SIMULATED — a simulated log $\neq$ an independent reproduction.

10. Open residuals — the families, each its own row

Grouped, never collapsed. Attack order: the form-leg and measure-leg targets first, then the which-outcomes leg and the cascade.

Measure leg (Gap-15 / A2): - Hole A — A2 measure uniqueness (highest leverage). Prove the maximal-symmetry chamber selector is the unique measure on $C=[1/2,3/2]^3$, target-blind — or refute it. OPEN. - Hole B — residual point-mass input. The $\delta$ at $(1,1,1)$, disclosed-not-derived; dissolves exactly when Hole A closes — no independent leverage. OPEN / disclosed. - Hole C — inter-sector weights, no clean invariant. Find a clean K₆ geometric invariant fixing the convex weights, or prove none exists. OPEN.

Form leg (Gap-14 / A1): - Hole D — derive non-contextuality (highest leverage on the form leg). Derive non-contextuality (hence $\mathrm{Tr}(\rho E)$) from the framework's own structure, without the forbidden import. OPEN. - Hole E — global operator-algebra type import-forbidden. The vN type A1 rides is import-forbidden; $Z(\mathcal A)$ uncomputed, $\xi_{R4}$ UNKNOWN (I2-DANGER-OPEN). OPEN / import-forbidden.

Which-outcomes leg + cascade + seam: - Hole F — the $a_6$ cascade onto Gap-01 (necessary-not-sufficient). Compute the signed total $\mathrm{tr}[a_6]$ at $d=13$ (bulk + ℤ₂ orbifold-defect). OPEN (upstream). - Hole G — system–bath split + pointer-basis falsifier. Run the partition, exhibit the pointer basis, run S3.b. OPEN. - Hole H — the Gap-14/Gap-15 seam. State whether the Born weights coincide with Gap-15's selector $\mu$, or disclaim. OPEN / silent. - Hole I — the $|\psi|^2$ weight derivation / recorded non-claim. Either a non-circular derivation, or a graded principled non-claim. OPEN.

None of these is closed by the §6.1 separation or the §6.2 localization.

11. Anti-claims (what this page refuses to say)

• The Born gate does not derive the rule. Formally: the measured floor $p(E)=\mathrm{Tr}(\rho E)$ is anchored, not derived; ANCHORED ≠ DERIVED.
• The BRST axioms do not force non-contextuality — the §6.1 countermodel proves the opposite: gauge-invariant, orbit-equivariant, contextual valuations exist.
• Leg A1 is anchored, not derived. The superseded per-leg label was AXIOM-OPEN / import-forbidden; the ratified closure-of-record grade is CERTIFIED-IRREDUCIBLE · RESOLVED +0 on the named posit BORN-A1 — a reduction onto one named, value-free, certified-irreducible posit, never a derivation of the functional form. ANCHORED ≠ DERIVED stays binding.
• Non-contextuality is a gluing condition, not a BRST output. $(G1)+(G2)$ supply the constraints, not the cross-context gluing $a=a'$.
• The chamber selector is not THE unique measure. Maximal symmetry does not single out one measure; the $S_3$-fixed locus is a line, and selecting a point is an extra input. Distinguished $\neq$ unique; existence $\neq$ uniqueness.
• Additivity does not fix the inter-sector weights. Gleason / Bunce–Wright is structurally silent across sectors.
• CH-2 has no asserted rate — the channel honest-halts; the bulk $a_6$ value is not the channel input.
• The cost-floor lever does not derive Born — it is recorded wrong-shape, an honest negative, not a hidden key.
• The measurement problem is not solved — explicitly out of scope, untouched even by a clean A2 measure.
• The frozen-branch hashes are audit anchors; they do not validate the physics. A simulated log $\neq$ an independent reproduction.

12. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package. Every one carries the κ³/π falsification test: a structure reverse-engineered to make Born appear is true-by-construction and relocates rather than closes.

1. Hole A (A2 uniqueness) — either (i) exhibit a larger compact symmetry group, genuinely forced by the frozen geometry, acting transitively/ergodically on $C$ (its unique Haar measure then closes A2; success criterion: uniqueness proven before and independently of any Born value), or (ii) prove no framework-forced symmetry acts ergodically — a valid terminal close certifying BORN-A2 irreducible. Machinery: homogeneous-space uniqueness; Weyl-chamber / Borel–de Siebenthal structure of K₆; the chamber data ($C$, $S_3$, $(1,1,1)$). Falsifier: the second outcome is as publishable as the first.
2. Hole B (point-mass) — no separate criterion; closes exactly when Hole A's measure is earned.
3. Hole C (inter-sector invariant) — exhibit a clean K₆ root-system / flag-manifold curvature invariant fixing the weights target-blind, or prove none exists.
4. Hole D (derive non-contextuality) — show the K₆ admissibility-chamber operator algebra forces a non-contextual assignment, recovering $\mathrm{Tr}(\rho E)$, with the algebra type derived from the geometry — without the forbidden import and passing the falsification test. Refuting result (valid close): prove no framework-internal structure forces the cross-context gluing — certifies BORN-A1 a genuine imported posit, the honest floor. Closing D dissolves Hole E in one move.
5. Hole E (algebra type) — compute $d_3\!\cdot\!u_2$ for $B(SU(3)\to PSU(3))$ (finite linear algebra) and run the $d_5=\beta_{P^1}$ spin-c Atiyah–Hirzebruch spectral sequence to pin $\xi_{R4}$, removing the I2-DANGER ambiguity. $\xi_{R4}$ is a shared R4 thread (Born, Gap-02 mass-gap, SG-4): computing it advances three gates. It must be computed, not assumed trivial.
6. Hole F ($a_6$ cascade) — reconcile the two K₆ Riemann routes (naturally-reductive $|{\rm Riem}|^2/|{\rm Ric}|^2=62/49\approx 1.2653$ vs FD-canonical $\approx 1.84$) by a third independent curvature computation; success criterion: the third method pins one value, resolving the $\sim 45.4\%$ scale-free gap. Necessary-not-sufficient for Born. Do not read the bulk value as the channel input; do not fabricate the defect or the rate.
7. Hole G (system–bath / pointer) — direct physics on $M_{3,1}\times K_6\times S^2\times S^1_Y/\mathbb Z_2$ ($D=13$); the only tractable slice short of that is a structural pre-flight stating the consistency conditions, shipped fail-closed as "ATTEMPT — NOT A CLOSURE." Do not decide the falsifier's outcome.
8. Hole H (the seam) — pose the weights-$\mu$ vs selector-$\mu$ relation precisely (documentary), defer the ruling to the cross-gap owner.
9. Hole I (weight derivation / non-claim) — a genuine non-circular derivation (the field's century-old prize), or a recorded principled non-claim with grade and falsifier; the latter is a documentary closure of the posture, not of the rule.

Closing Holes A + D earns both legs (still given the measured floor); the rest harden the which-outcomes leg and the cascade.

13. The dissolved unicorns — shared ceilings, not private gaps

Three statements are universal negatives over all of mathematics / all of physics — limits on all knowledge, never claimed as proven, never listed as a defect of this program: - No interpretation in any mathematical framework could ever derive the Born rule without importing a rule-strength assumption. The decades-open community fact (every known route relocates) is the ceiling. The bounded contribution is to name the relocation precisely and prove (countermodel) that this framework's gauge axioms in particular do not force it. - The constructed A2 chamber measure is THE absolutely unique measure over every conceivable measure on the state space. Uniqueness-against-all-alternatives is unprovable in principle; the bounded target is uniqueness within the framework's own declared structure. - The full interpretive measurement problem is solved. Explicitly scope-excluded — an open-ended foundational question for the whole field, not a finite plug-able hole.

14. Completion tests for this page

Required presence (all met): gate roll-up CERTIFIED-IRREDUCIBLE · RESOLVED +0 with the historical OPEN roll-up shown as history · Leg A1 named value-free posit (superseded per-leg label AXIOM-OPEN / import-forbidden) · Leg A2 uniqueness-fails residual shown · the A1 separation DERIVED with diagnostic ($a\neq a'$ survives a non-gauge rotation) · the measured floor ANCHOR-BORN-QUANTUM-KINEMATICS MEASURED-ANCHOR · $E$ / quantum-kinematics not derived · frozen hashes AUDIT ONLY · $(G1)$, $(G2)$ · Gleason / Busch / Bunce–Wright table · no-I₂ / $\xi_{R4}$ I2-DANGER-OPEN · chamber $C=[1/2,3/2]^3$, $W=S_3$, witness $(1,1,1)$ · existence-yes / uniqueness-no · point-mass disclosed · inter-sector silence · CH-2 honest-halt · the duality (two axioms) split · κ³/π falsification test · every Hole A–I as its own residual · the gate's anti-claims · the three dissolved unicorns.

Required absence (all held): no claim that Born is derived/physics-closed · A1 derived (an anchored reduction is never a derivation) · non-contextuality forced by BRST · the chamber selector is THE unique measure · $(1,1,1)$ is the unique measure · CH-2 has a computed rate · the bulk $a_6$ is the channel input · the cost floor derives Born · the measurement problem solved · hashes validate physics · a simulated log is a reproduction · any open residual asserted proven/computed/closed · any reader-visible build-process vocabulary.

Completion report. Tests passed: all required-presence and required-absence items above. Tests failed: none. Open items: Holes A–I (the gate's named residuals). Assumptions made: none beyond the frozen branch and the single measured floor — nothing here upgrades the gate's grade.

Binding closing statement. The Born rule (Gap-14 / Gap-15) stands, on the ratified board (2026-07-08), at CERTIFIED-IRREDUCIBLE (REDUCED-TO-AXIOM on the named posit BORN-A1) · RESOLVED +0 — the exponent p = 2 reproduced exactly (Gleason-forced once non-contextuality is granted), no free numerical parameters fit. Residuals shown, never hidden: Leg A1 rides an imported, import-forbidden algebra type as a named, value-free posit — the functional form is anchored, not derived; Leg A2 carries uniqueness-fails (point-mass disclosed, no clean inter-sector invariant). The genuine, non-promoting wins are the A1 separation countermodel (real, as a no-go) and the K₆ chamber-selector sharpening of A2 (real, as reductive work). Ceiling: a reached certified-irreducible terminal — a serious candidate, NOT experimentally validated. Frozen branch dcc66f1b2685 / a5b1e6f9d951 READ-ONLY; selection ≠ derivation · given-E ≠ derivation-of-E · frozen/reproducible ≠ proven-unique · ANCHORED ≠ DERIVED · AXIOM-CLOSED ≠ atomic · sharpened ≠ derived.

This gate anchor ledger follows the canonical eleven-part shape and universal table of the SG-4 ledger.

See also: the anchoring method · A0 — the master anchor · Layer 2 — no unpaid exact labels (why additivity is a filter, not a selector) · Layer 4 — carrier-forcing & the given-E wall (why the geometry is load-bearing but not certifying) · SG-4 — Hypercharge & anomaly (the canonical gate ledger) · the full Born dossier.